markowitz's portfolio selection model and related, Lecture notes of Financial Economics

Download markowitz's portfolio selection model and related and more Lecture notes Financial Economics in PDF only on Docsity! MARKOWITZ’S PORTFOLIO SELECTION MODEL AND RELATED PROBLEMS by ABHIJIT RAVIPATI A thesis submitted to the Graduate School-New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Master of Science Graduate Program in Operations Research written under the direction of Prof. Dr. ANDRAS PREKOPA and approved by ________________________ ________________________ ________________________ New Brunswick, New Jersey January 2012 ii ABSTRACT OF THE THESIS MARKOWITZ‟S PORTFOLIO SELECTION MODEL AND RELATED PROBLEMS By ABHIJIT RAVIPATI Thesis Director: Prof Dr. ANDRAS PREKOPA Markowitz's portfolio selection theory is one of the pillars of theoretical finance. This formulation has an inherent instability once the mean and variance are replaced by their sample counterparts. The problem is amplified when the number of assets is large and the sample covariance is singular or nearly singular. This poses a fundamental problem, because solutions that are not stable under sample fluctuations may look optimal for a given sample, but are, in effect, very far from optimal with respect to the average risk. The paper starts with a general introduction to Markowitz‟s portfolio theory and then discusses further developments and a few notable works in the area and later moves on to discuss the need for regularization and points out a few existing methods for regularization. After which a formulation of the optimal portfolio selection is presented and ends with a few numerical examples. 1 Markowitz’s Portfolio Theory 1.1 Introduction A little over forty years ago, a University of Chicago graduate student in economics, while in search of a dissertation topic, ran into a stockbroker who suggested that he study the stock market. Harry Markowitz took that advice and developed a theory that became a foundation of financial economics and revolutionized investment practice. His work earned him a share of 1990 Nobel Prize in Economics. A basic premise of economics is that, due to the scarcity of resources, all economic decisions are made in the face of trade- offs. Markowitz identified the trade-off facing the investor: risk versus expected return. The investment decision is not merely which securities to own, but how to divide the investor's wealth amongst securities. This is the problem of “Portfolio Selection;” hence the title of Markowitz‟s seminal article published in the March 1952 issue of the Journal of Finance. In that article and subsequent works, Markowitz extends the techniques of linear programming to develop the critical line algorithm. The critical line algorithm identifies all feasible portfolios that minimize risk (as measured by variance or standard deviation) for a given level of expected return and maximize expected return for a given level of risk. When graphed in standard deviation versus expected return space, these portfolios form the efficient frontier. The efficient frontier represents the trade-off between risk and expected return faced by an investor when forming his portfolio. Most of the efficient frontier represents well diversified portfolios. This is because diversification is a powerful means of achieving risk reduction. 2 Markowitz developed mean-variance analysis in the context of selecting a portfolio of common stocks. Over the last two decades, mean-variance analysis has been increasingly applied to asset allocation. Asset allocation is the selection of a portfolio of investments where each component is an asset class rather than an individual security. In many respects, asset allocation is a more suitable application of mean-variance analysis than is stock portfolio selection. Mean-variance analysis requires not only knowledge of the expected return and standard deviation on each asset, but also the correlation of returns for each and every pair of assets. Whereas a stock portfolio selection problem might involve hundred of stocks (and hence thousands of correlations), an asset allocation problem typically involves a handful of asset classes (for example stocks, bonds, cash, real estate, and gold). Furthermore, the opportunity to reduce total portfolio risk comes from the lack of correlation across assets. Since stocks generally move together, the benefits of diversification within a stock portfolio are limited. In contrast, the correlation across asset classes is usually low and in some cases negative. Hence, mean-variance is a powerful tool in asset allocation for uncovering large risk reduction opportunities through diversification. 1.2 Assumptions As with any model, it is important to understand the assumptions of mean-variance analysis in order to use it effectively. First of all, mean-variance analysis is based on a single period model of investment. At the beginning of the period, the investor allocates his wealth among various asset classes, assigning a nonnegative weight to each asset. 3 During the period, each asset generates a random rate of return so that at the end of the period, his wealth has been changed by the weighted average of the returns. In selecting asset weights, the investor faces a set of linear constraints, one of which is that the weights must sum to one. Based on the game theory work of Von Neumann and Morgenstern, economic theory postulates that individuals make decisions under uncertainty by maximizing the expected value of an increasing concave utility function of consumption. In a one period model, consumption is end of period wealth. In general, maximizing expected utility of ending period wealth by choosing portfolio weights is a complicated stochastic nonlinear programming problem. To summarize the assumptions: 1. Investors seek to maximize the expected return of total wealth. 2. All investors have the same expected single period investment horizon. 3. All investors are risk-adverse, that is they will only accept greater risk if they are compensated with a higher expected return. 4. Investors base their investment decisions on the expected return and risk. 5. All markets are perfectly efficient (e.g. no taxes and no transaction costs). The utility function is assumed to be increasing and concave. In terms of the approximating utility function, this translates into expected utility being increasing in expected return (more is better than less) and decreasing in variance (the less risk the 6 If the investor has access to risk free investment, the risk of the portfolio can further be reduced, which is shown in Fig.1 by the CAL. The risk-free asset is the (hypothetical) asset which pays a risk-free rate. In practice, short-term government securities (such as US treasury bills) are used as a risk-free asset, because they pay a fixed rate of interest and have exceptionally low default risk. The risk-free asset has zero variance in returns (hence is risk-free); it is also uncorrelated with any other asset (by definition, since its variance is zero). Within the application of portfolio theory the following two quantities will need to use the corresponding units of measurement throughout the computation: 1. Historical Values: This is the source data which is given in absolute or relative terms. 2. Expected Returns: The expected return of the investment over the period considered which should be given and will be returned in the units used (i.e. absolute or relative) by the historical values. The values of the expected return which are either evaluated or given will be or will need to be in accordance with the units used within the historical values. “ To use the E-V rule in the selection securities we must have procedures for finding reasonable µi and ρij. These procedures I believe, should combine statistical techniques and the judgment of practical knowledge…………. One suggestion as to tentative µi, ρij is to use observed for some period of the past. I believe, better methods, which take into account more information can be found. I believe that what is needed is essentially a 7 “probabilistic” reformulation of security analysis.” - Markowitz, H.M. (March 1952). "Portfolio Selection". The Journal of Finance 1.4 Later Developments Markowitz‟s selection model is fundamental to the foundation of the current theory of asset allocation. Since Markowitz proposed his model, numerous portfolio selection models have been developed to advance the former and portfolio theory has been improved and completed in several directions. Some models have been developed to minimize semivariance in different cases such as Huang [15] and Markowitz [16], while other researchers like Konno and Suzuki [17], Liu, Wang and Qiu [18] and Pornchai, Krushnan, Shatid and Arun [19] added the skewness in consideration for portfolio selection. The common assumptions are that the investor has enough historical data and that the situation of asset markets in future can be correctly predicted by the historical data. Since sometimes this is not, practical problems arise. For example, when new stocks are listed in the stock market, there is no historical information for these securities. Random, fuzzy and random fuzzy optimization models proved some useful methods for investors to tackle the uncertainty. A number of researchers have shown that mean-variance efficient portfolios, based on estimates, are highly sensitive to perturbations of these estimates. Jobson, Korkie and Ratti (20) and Jobson and Korkie (21) detail these problems and suggest the use of shrinkage estimators. 8 Some authors like Carlsson, Fuller and Majlender [22], Leon, Liern and Vercher [23] and Vercher, Bermudez and Segura [24] use fuzzy numbers to replace uncertain returns of the securities. Tanaka and Guo [25] and Tanaka, Guo and TÄurksen [26] used possibilistic distributions to model uncertainty in returns. Arenas-Parra, Bilbao-Terol and Rodrguez-Ura [27] introduced vague goals for return rate, risk and liquidity based on expected intervals. A measure of downside risk is incorporated by Feiring, Wong, Poon, and Chan (28), and Konno, Shirakawa, and Yamazaki (29) who use an approximation to the lower semi-third moment in their Mean-Absolute Deviation-Skewness portfolio model. Konno and Yamazaki proposed the mean absolute deviation (MAD) model as an alternative to the mean variance (MV) model claiming that it retains all the positive features of the mean variance model , not only saves computing time but also does not require the covariance matrix. Addressing the issue of estimation risk, Frost and Savarino (30) show that constraining portfolio weights, by restricting the action space during the optimization, reduces estimation error. Jorion (31) proposes a resampling method aimed at estimation error. In an attempt to maintain the decision simplicity associated with the efficient frontier and still accommodate parameter uncertainty, Michaud (32) proposes a sampling based method for estimating a resampled efficient frontier. Polson and Tew (33) argue for the use of posterior predictive moments instead of point estimates for mean and variance of an assumed sampling model. Using a Bayesian approach, Britten-Jones (34) proposes placing informative prior densities directly on the portfolio weights. Chopra and Ziemba (35) showed that errors in 11 Investor‟s perception: A large number of investors were not fully convinced of the validity of the standard deviation as a measure of risk. Transaction costs and cut-off effect: This means that the investor who invests in many different stocks in small costs will be inconvenienced as he will have to bear the burden of transaction costs. Also, since the investor cannot buy stocks in fractions and he/she will have to round it off to integers. 1.4.1.2 Model They introduced the L1 risk function w(x) = E[ Where, = Random variable representing the rate of return on asset Sj = Amount invested in Sj = Total fund amount E [.] = Expected value of random variable in bracket They then go on to state and prove the following theorem: If (R1,……..Rn) are multivariate normally distributed , then w(x) = Where = Standard deviation 12 They proved that these two measures (w(x) and Ri) are the same if (R1…Rn) are multivariate normally distributed. So the Model becomes the following; Min w(x) E[ ST = , ………………… 12.1 0 , j= 1,…….,n . Konno and Yamazaki assumed that the expected value of the random variable can be approximated by the average from the data. Therefore, = Now, E[ = Let j= 1,…….,n; t=1,…..,T . Model in 12.1 can be stated as, 13 Min / T ST = , 0 , j= 1,…….,n Which is equivalent to the following linear program: Min t= 1,…..,T, t= 1,…..,T, ST = , 0 , j= 1,…….,n Konno-Yamazaki state the following advantages over Markowitz‟s model: 1. No need to calculate the covariance matrix. 2. Solving their linear program is much easier compared to solving a quadratic program. 3. The optimal solution size is smaller 4. T can be uses as a control variable to restrict the number of assets in the portfolio 16 Thus, Mp represents the portfolio‟s minimum return at the end of each time period and since Mp is being maximized, the portfolio will take on the maximum value of the minimum returns. According to Young, this model presents logical advantages over other portfolio optimization models if asset prices are not normally distributed and similar results when they are. He states an equivalent model that seeks to maximize expected return, subject to a restriction that the portfolio return exceeds some threshold H in each observation period: t = 1,…..,T, ST j=1,….,N. This model has considerable advantage as it is a linear program and it allows the model to treat additional complexities such as: 1. Transaction costs 2. Logical side constraints like  Inclusion/exclusion of both assets a and b  Holding more than $d worth of asset a Thus the minimax model is capable of incorporating a large number of modeling complexities and variations. 17 1.4.3 Black and Litterman, 1992 1.4.3.1 Introduction The Black-Litterman asset allocation model, created by Fischer Black and Robert Litterman, is a portfolio construction method that overcomes the problem of highly concentrated portfolios, input-sensitivity, and estimation error maximization. Their model uses a Bayesian approach to combine the subjective views of an investor regarding the expected returns of one or more assets with the market equilibrium vector of expected returns (the prior distribution) to form a new, mixed estimate of expected returns. The Black-Litterman asset allocation model was introduced in Black and Litterman (1990) and expanded in Black and Litterman (1991, 1992). The Black Litterman model combines the CAPM (Sharpe (1964)), reverse optimization (Sharpe (1974)), mixed estimation (Theil (1971, 1978)), the universal hedge ratio / Black‟s global CAPM (see Black (1989a, 1989b)), and mean-variance optimization (Markowitz (1952)). The approach works by combing historical information with additional data and forms the updated distribution of expected returns. If the investor has no subjective opinions the weights are based on market equilibrium data, but if he does hold any subjective views then the weights on individual assets are shifted from the market equilibrium weights. The key inputs to the Black and Litterman model are market equilibrium returns and investor views. This framework incorporates the investor views that helps investors control the magnitude of the tilts caused by views. 18 1.4.3.2 Market equilibrium returns The Black and Litterman model uses the market equilibrium weights or capital asset pricing model (CAPM) as the basis. CAPM is developed by forming the efficient frontier of the market portfolios and tracing the capital market line (CML). The CML is tangent to the efficient frontier at the market portfolio therefore, there is no other combination of risky and riskless assets that can provide better returns for a given level of risk. CAPM: E ( = + ( - = + ( - Where, E ( = Expected return on asset i = Risk free asset return = Return on market portfolio = Standard deviation of returns on asset i = Standard deviation of returns on market portfolio = 21 1.4.3.4 Black- Litterman equation The Black-Litterman equation combines the equilibrium returns and investor views into one equation to determine expected returns which is used to determine the optimal portfolio weights. E[R] = [ ………….21.1 Where, E [R] = Vector of combined results (N,1) = Scalar indicating uncertainty of CAPM prior = Covariance matrix of equilibrium excess returns (N,N) P = Matrix of investor views (N,1) = Diagonal covariance matrix of view error terms (K,N) = Vector of equilibrium excess returns (N,1) Q = Vector of investor views (K,1) In case the investor is unconstrained we use 19.1 to determine optimal portfolio weights by, …….………21.2 22 Where is the vector derived from 21.1. 21.2 is the solution to the unconstrained optimization problem Overall, the Black-Litterman model overcomes the most-often cited weaknesses of mean- variance optimization such as highly concentrated portfolios, input-sensitivity, and estimation error-maximization, helping users to realize the benefits of the Markowitz paradigm. Regularization 2.1 Introduction In the original formulation of the Markowitz portfolio theory, the underlying process was assumed to be multivariate normal. Accordingly, reward was measured in terms of the expected return, risk in terms of the variance of the portfolio. Unfortunately, the nature of portfolio selection is not compatible with this limit. Institutional portfolios are large, with N's in the range of hundreds or thousands, while considerations of transaction costs and non-stationarity limit the number of available data points to a couple of hundreds at most. Therefore, portfolio selection works in a region, where size of the portfolio and sample size (i.e. size of the time series available for each asset) are, at best, of the same order of magnitude. This, however, is not the realm of classical statistical methods. It is evident that portfolio theory struggles with the same fundamental difficulty that is underlying basically every complex modeling and 23 optimization task: the high number of dimensions and the insufficient amount of information available about the system. In order to solve the mean variance problem, the expected return and the covariance matrix of the vector of security return, which are unknown, need to be estimated from available data set. In particular, an estimate of the inverse of the covariance matrix is needed. The sample covariance often used in practice may be the worst choice because it is typically nearly singular, and sometimes not even invertible. Markowitz mean-variance rule can be viewed as a trade-off between the expected return and the variance of the returns. The mean variance problem consists of choosing the vector x , to minimize the variance of the resulting portfolio rp,t+1=x‟rt+1, where rt+1 is the excess return Rt+1 – Rt f , for a pre- determined target expected return of the portfolio µp : Minx Var[rp] = w‟Cw S.T. E[rp] =w‟µ = µp The optimal portfolio is given by, w* = C -1 µ Where µ = Conditional means, C = Covariance matrix. The issue of ill-conditioned covariance matrices is important because inverting such a matrix increases dramatically the estimation error and then makes the mean variance solution unreliable. 26 If one replaces the expectation by sample average µ, the problem becomes : x* = argminx 2 2 S.T. x‟µ = µp As mentioned before, the solution of this problem may be very unreliable if RR‟ is nearly singular. To avoid having explosive solutions, we can penalize the large values by introducing a penalty term applied to a norm of x. Depending on the norm we choose, we end up with different regularization techniques. Xi 2.3.1 Bridge method For Υ > 0 the Bridge estimate is given by xт* = argminx 2 2 + т xi Υ Where т is the penalty term. The Bridge method includes two special cases. For Υ = 1 we have Lasso regularization, while Υ = 2 leads to the Ridge method. The term xi Υ can be interpreted as a transaction cost. It is linear for Lasso, but quadratic for the ridge. 27 2.3.2 Least Absolute Shrinkage and Selection Operator (LASSO) The Lasso regularized solution is obtained by solving : xт* = argminx 2 2 + т 1 S.T. x‟ µ = µp For two differently penalty constants т1 and т2 the optimal regularized portfolio satisfies : (т1 - т2) ( x [т 2 ] 1 - x [т 1 ] 1) >=0 then the higher the l1 – penalty constant (т), the sparser the optimal weights. So that a portfolio with non-negative entries corresponds to the largest values of т and thus to the sparsest solution. In particular the same solution can be obtained for all т greater than some value т0. Brodie et al. consider models without a risk free asset. Using the fact that all the wealth as invested (x‟1N = 1), they use the equivalent formulation for the objective function as : µp1T – Rx 2 2 + 2т xi + т which is equivalently to a penalty on the short positions. The Lasso regression then regulates the amount of shortening in the portfolio designed by the optimization process, so that the problem stabilizes. The general from of the l1- penalized regression with linear constraints is: xт * = argminx∊H b – Ax 2 2 + т x 1 28 H is an affine subspace defined by linear constraints. The regularized optimal can be found using an adaptaion of the homotopy / LARS algorithm as described in Brodie et al. . 31 Substituting the above equations in the KKT necessary conditions, c – 2Cx + A T λ + - A T λ - + v = 0 Now, the problem is to find λ, x, V which satisfy the following formulation, Min ui Ax = b S.T. – 2Cx + A T λ + - A T λ - + v + Fu = -c x, λ + , λ - ,v, u >= 0 Where F is a diagonal (n x n) matrix F(i,i)= 1 , if c(i)>=0 -1 , if c(i) < 0 Example The above stated formulation has been tested using the following data: A= [-1 1 1 -1 1] b= [1] 32 C= c= Running the code in appendix 1 the following solution is arrived upon as optimal: X= λ + = λ - = V= 33 u= Appendix Matlab Code [n1, n1] = size(C1); I=eye(n1); Z1=zeros(1,2*n1+2); for i=1:n1 for j=1:n1 if ((i==j) & ( c1(i) >= 0)) F(i,j)=1; elseif ((i==j) & ( c1(i)< 0)) F(i,j)=-1; else F(i,j)=0; end end 36 fprintf('Number of iterations is %d\n',iter); x = zeros(n,1); x(B) = xB; fprintf('Optimal objective value is %f\n',c'*x); obj = c'*x; return; end; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Step 3 :- Solve A_Bd = a_{N(k)} % Find theta = Min_{i=1,...,m|d_i > 0} xB(i)/d(i) % Let theta = xB(l)/d(l) % x_{B(l)} is the leaving basic variable % Also check for unboundedness if d <= 0 d = A(:,B)\A(:,N(k)); zz = find(d > eps)'; if (isempty(zz)) error('System is unbounded\n'); 37 end [qw,er]= ismember(N(k),1:n1); [qw1,as]= ismember(N(k) + n1+2,B); [qw2,ty]= ismember(N(k),n1+3:2*n1+2); [qw1,df]= ismember(N(k) - n1-2,B); if er~=0 & as~=0 l= find(B== (N(k) +n1+2)); theta = xB(l)/d(l); elseif ty~= 0 & df~=0; l= find(B== (N(k) -n1-2)); theta = xB(l)/d(l); else [theta,ii] = min(xB(zz)./d(zz)); l= zz(ii(1)); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Step 4:- Update B and N % Also x(B(i)) = x(B(i)) - theta*d(i), i=1,...,m and i not equal to l 38 % x(B(l)) = theta temp = B(l); B(l) = N(k); N(k) = temp; xB = A(:,B)\b; end; % while 41 [20] J.D. 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